An open question of Corteel, Lovejoy and Mallet
Kanade, Shashank and Kurşungöz, Kağan and Russell, Matthew (2017) An open question of Corteel, Lovejoy and Mallet. In: Number Theory: In Honor of Krishna Alladi's 60th Birthday, University of Florida, Gainesville, FL, US (Accepted/In Press)
Corteel, Lovejoy and Mallet concluded their paper \An extension to overpartitions of the Rogers-Ramanujan identities for even moduli" with an open question of investigating the combinatorial properties of a q-series with two additional parameters.We settle their question, unfortunately in the negative, by showing that the series yields only the known results in overpartitions. However; when one annihilates one of the parameters, the resulting series have nice integer partitions interpretations. Those series appeared in another publication as well. In particular, Corteel, Lovejoy, and Mallet's series involve an index d. This index unifies two classes of overpartition identities for d = 1 and d = 2, but does not give additional overpartition identities for d => 3. Upon setting one of the parameters zero, one does get regular partition identities for all d. The proofs are conventional, formal verifications for brevity, but we show how to make the proofs constructive.
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